The application of non-uniform magnetic field for thermal enhancement of the nanofluid flow inside the U-turn pipe at solar collectors

The improvement of heat transfer inside the solar heat exchangers is important for the development of solar energy in an urban area. In this study, the usage of a non-uniform magnetic field on the thermal efficiency of the nanofluid (Fe3O4) streaming inside the U-turn pipe of solar heat exchangers is examined. Computational fluid dynamic is applied to visualize the nanofluid flow inside the solar heat exchanger. The role of magnetic intensity and Reynolds number on thermal efficiency are fully investigated. The effect of single and triple sources of the magnetic field is also studied in our research. Obtained results indicate that the usage of the magnetic field results in the production of vortex in the base fluid and heat transfer improves inside the domain. Our finding indicates that the usage of the magnetic field with Mn = 25 K would improve the average heat transfer by about 21% along the U-turn pipe of solar heat exchangers.

www.nature.com/scientificreports/ i.e. boiling and melting since this would change the thermal properties of the process [28][29][30] . Although, the usage of a magnetic source either uniform or non-uniform near the nanofluid with ferro particles has been explored in current research, this aspect of the nanofluid flow was not comprehensively explored in different sections of the heat exchangers [31][32][33] . In most of these studies, a theoretical approach is used for the thermal analysis of the nanofluid flow [34][35][36] . The numerical technique of Computational Fluid dynamics is also used for the investigation of the heat transfer of heat exchangers 37,38 . Due to the low cost of computational investigations, this technique is considered as the initial method for the pre-evaluation of new innovative approaches for the development of current research 39,40 . Although several investigations have focused on the uniform-magnetic field for the improvement of heat exchangers, the non-uniform magnetic field was studied in limited articles via computational fluid dynamics [41][42][43] .
In this article, the role of a non-uniform magnetic field on the thermal improvement of nanofluid streaming along a U-turn double pipe heat exchanger (Fig. 1). The flow characteristics and heat properties of the nanofluid Figure 1. The schematic of U-turn pipe in presence of (a) single (b) triple non-uniform magnetic source. www.nature.com/scientificreports/ flow are examined and analyzed by computational fluid dynamics. The streamline of base flow with ferro particles in a two-dimensional model is fully explored to reveal the main flow factors which improve the thermal aspects of the nanofluid. Impacts of nanofluid Reynolds number and magnetic intensities are also simulated on the hydrodynamic of the nanofluid stream. In addition, variation of the heat transfer along the U-turn pipe is plotted in different conditions.

Governing equation and computational methodology
Nanofluid is the mixture of ferro particles (3-15 nm) inside the base fluid which is water in this study. The simulation of the nanofluid is done while the base fluid is incompressible water with the thermal characteristics of ferrofluid. It is assumed that the nanofluid stream is steady, incompressible, and laminar [44][45][46] . The main governing equations of the 2D nanofluid stream with mentioned assumption are as follows: There are two source terms in the momentum equations and they are associated with magnetic field and known as kelvin Force. These source terms are calculated via these formulas: where the value of M is calculated via these equations: The non-dimensional value for evaluation of the magnetic intensity is calculated via In proceeding published articles, full details of the applied non-uniform magnetic field on the main governing equations of the ferrofluid are presented and explained. The thermo-characteristics of the base fluid, air, and water are presented in Table 1.
The density, viscosity, and heat capacity of the mixture (water with ferro particles) are calculated via www.nature.com/scientificreports/ To compare hydro and thermal characteristics of nanofluid stream, Reynolds and Nusselt numbers should be calculated as follows: The selected model of a U-turn double pipe with the nanofluid flow is displayed in Fig. 1. The sources of the magnetic field are demonstrated with the non-uniform distribution. Uniform heat flux (1000 W/m 2 K) is applied on the wall as depicted in the figure. It is assumed that the ferrofluid with constant temperature (300 K) and velocity enters the pipe. The size of the pipe is d/D = 0.8.
A grid study is also done for the numerical simulation of the ferrofluid inside the pipe. Figure 2 illustrated produced grid for chosen U-turn model. The generated grid is fully structured and the size of the grid near the wall is less than other regions due to the interaction of the magnetic field on the nanofluid stream. Grid independency analysis is done ( Table 2) to attain the optimum grid cells for the introduced geometry. The results of grid independency are presented in Fig. 3. As noticed in the table, the size of grids is changed and its effects on the thermal characteristic of the single pipe are compared. The model with a grid number of 9000 cells (30 × 300) is chosen for this study.

Results and discussion
The validation of our method are done for a nanofluid stream along a single pipe and our data are compared with both experimental and computational data [46][47][48][49] . As demonstrated in Fig. 3, the heat transfer coefficient of nanofluid with two nanoparticle volume fractions at Re = 1500 is simulated and compared results confirmed that the deviation of our result is less than 8% which is acceptable for our future studies [50][51][52] . This approach is used in different scientific problems [53][54][55][56][57][58] . Figure 4 illustrates the impacts of the magnetic source on streamlining of nanofluid flow inside the U-turn pipe. When the single magnetic source is applied in the vicinity of the mid-section of the U-turn, one single vortex is produced. The formation of this vortex is the main result of the kelvin force. In the case with three magnetic sources, there are three vortices inside the domain. It is found that the first vortices (wire 3) are larger than the others. The main reason for the production of vortices is the non-uniformity of the magnetic field. How the formation of the vortices improves the heat transfer would be explained in the next sections.
The influence of these vortices on the heat transfer along the U-turn pipe is illustrated in Fig. 5. In this figure, the fluctuation of the Nusselt number shows the influence of the magnetic source on the heat transfer of the U-turn pipe in different Reynolds numbers. As the velocity of inlet nanofluid flow increases, the impacts  www.nature.com/scientificreports/ of magnetic source on the hydrodynamic of nanofluid flow are limited due to the high momentum of the fluid. Thus, the size of the vortex is restricted, and consequently, the heat transfer rate decrease as demonstrated in Fig. 5. One of the main aspects of these vortices is the high-velocity gradient inside the domain which makes the nanofluid stay more. Figure 6 demonstrates the heat transfer along the U-turn pipe in different Reynolds numbers without a magnetic field. Effects of the magnetic field on the temperature variation of nanofluid flow are depicted in Fig. 7. It is observed that the temperature change occurs in the region where the vortex is produced. The variation of the temperature is not only related to the vortices but also associated with the changes in heat capacity of nanofluid under impacts of non-homogeny magnetic force. The non-dimensional temperature contour demonstrates these effects in our model. The intensity of the magnetic field also intensifies the formation of strong vortices and consequently, heat transfer is improved.
A comparison of heat transfer for the single and triple non-uniform magnetic source is plotted in Fig. 8. The formation of the peak in the plot represents the formation of the vortices under the impact of the nonuniform magnetic field. The maximum Nusselt number occurs near wire 3 of the triple magnetic source case. The streamlined figure also confirms that a large vortex is produced in this area. In fact, this is due to the shape of the U-turn pipe which tends the flow to move in the far wall rather than the inner wall. Thus, there is less resistance to the disruption of the vortex. Figure 9 illustrates the temperature contour of a single and triple magnetic source as well as a model without a magnetic field. The temperature variation near the source of the magnetic field verifies that the existence of the non-uniform magnetic field improves the heat transfer on the U-turn pipe. As the number of the source of the magnetic field is increased, a larger area inside the domain is under the impact of the magnetic source.
The influence of magnetic intensity on the average Nusselt number along the U-turn pipe is presented in Fig. 10. The inlet velocity is constant (Re = 50) and the source of non-uniform magnetic fields is uniform. The variation of the average Nusselt number with changes in the magnetic source is almost linear. The average Nusselt  www.nature.com/scientificreports/ number increase by about 21% when the magnetic source with a magnetic intensity of Mn = 258,000 is applied by a single source. Following equation is obtained for estimation of the average heat transfer along U-turn pipe.

Conclusion
The influence of a non-uniform magnetic field on the heat transfer of nanofluid through a U-turn pipe is explored in the present study. CFD technique is used to model the hydrodynamic and thermal characteristics of the nanofluid with Fe203 ferro-particles under impacts of the single and triple magnetic field in a U-turn pipe. The role of magnetic intensity and inlet nanofluid velocity on the hydrodynamic of the nanofluid flow. The production of the vortices and its impact on the heat transfer of the based fluid are extensively explained. Temperature contour of various fluid conditions is also present in the current study. Obtained results show that the application of a (14) Nu avg = 0.0004Mn + 5.44     www.nature.com/scientificreports/